# Inverse Observability Inequalities for Integrodifferential Equations in Square Domains

## TL;DR

This paper establishes inverse observability inequalities for integrodifferential equations modeling viscoelastic membranes in square domains, using spectral analysis to account for memory effects in wave equations.

## Contribution

It introduces new spectral estimates for integrodifferential operators and proves inverse observability inequalities for viscoelastic membrane models.

## Key findings

- Spectral properties of integrodifferential operators are characterized accurately.
- Inverse observability inequalities are proven for viscoelastic membrane equations.
- Results extend control theory to systems with memory effects.

## Abstract

In this paper we will consider oscillations of square viscoelastic membranes by adding to the wave equation another term, which takes into account the memory. To this end, we will study a class of integrodifferential equations in square domains. By using accurate estimates of the spectral properties of the integrodifferential operator, we will prove an inverse observability inequality.